In Minkowski spacetime and Cartesian coordinates , the conservation equations (1, 2) can be written in vector form asi = 1 ,2 ,3. The state vector is defined by
In the non-relativistic limit (i.e., , ) , , and approach their Newtonian counterparts , , and , and Equations (5) reduce to the classical ones. In the relativistic case the equations of system (5) are strongly coupled via the Lorentz factor and the specific enthalpy, which gives rise to numerical complications (see Section 2.3).
In classical numerical hydrodynamics it is very easy to obtain from the conserved quantities (i.e., and ). In the relativistic case, however, the task to recover from is much more complicated. Moreover, as state-of-the-art SRHD codes are based on conservative schemes where the conserved quantities are advanced in time, it is necessary to compute the primitive variables from the conserved ones one (or even several) times per numerical cell and time step making this procedure a crucial ingredient of any algorithm (see Section 9.2).
© Max Planck Society and the author(s)